arXiv subjects

Jeremy Rouse

Publications and source records attributed to Jeremy Rouse.

At least 19 recordsLinked to original sources

arXiv2026

Rational points on modular curves via maps to elliptic curves with rank zero

A fundamental problem in arithmetic geometry is to determine the image of the mod $N$ Galois representation for all elliptic curves over $\mathbb{Q}$ and integers $N \geq 1$. For a given subgroup $G \le \mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})$, there is a modular curve $X_G$ whose rational points parametrize elliptic curves for which the image of the mod $N$ Galois representation is contained in $G$. If $X_G$ admits a map to an elliptic curve $E/\mathbb{Q}$ for which $E(\mathbb{Q})$ has rank $0$, then its rational points can be effectively determined, provided that a map $X_G \to E$ is known. In this article, we give a method for constructing such maps. Using this method, together with existing methods and results, we systematically determine the rational points of $X_G$ for more than $99\%$ of modular curves of level at most $70$.

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arXiv2025

Modular Forms with Only Nonnegative Coefficients

We study modular forms for $\textrm{SL}_2(\mathbb{Z})$ with no negative Fourier coefficients. Let $A(k)$ be the positive integer where if the first $A(k)$ Fourier coefficients of a modular form of weight $k$ for $\textrm{SL}_2(\mathbb{Z})$ are nonnegative, then all of its Fourier coefficients are nonnegative, so that $A(k)$ can be interpreted as a ``nonnegativity Sturm bound''. We give upper and lower bounds for $A(k)$, as well as an upper bound on the $n$th Fourier coefficient of any form with no negative Fourier coefficients.

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arXiv2025

A uniform bound on the smallest surjective prime of an elliptic curve

Let $E/\mathbb{Q}$ be an elliptic curve without complex multiplication. A well-known theorem of Serre asserts that the $\ell$-adic Galois representation $\rho_{E,\ell^\infty}$ is surjective for all but finitely many prime numbers $\ell$. Considerable work has gone into bounding the largest possible nonsurjective prime; a uniform bound of $37$ has been proposed but is yet unproven. We consider an opposing direction, proving that the smallest prime $\ell$ such that $\rho_{E,\ell^\infty}$ is surjective is at most $7$. Moreover, we completely classify all elliptic curves $E/\mathbb{Q}$ for which the smallest surjective prime is exactly $7$.

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arXiv2024

Near coincidences and nilpotent division fields

Let $E/\mathbb{Q}$ be an elliptic curve. We say that $E$ has a near coincidence of level $(n,m)$ if $m \mid n$ and $\mathbb{Q}(E[n]) = \mathbb{Q}(E[m],\zeta_{n})$. We classify near coincidences of prime power level and use this result to give a classification of values of $n$ for which ${\rm Gal}(\mathbb{Q}(E[n])/\mathbb{Q})$ is a nilpotent group. Along the way we prove a Gauss-Wantzel analog for the elliptic curve $E\colon y^2 = x^3-x$, showing that $\mathbb{Q}(E[n])/\mathbb{Q}$ is constructible if and only if $\varphi(n)$ is a power of 2. Assuming that there are no non-CM rational points on the modular curves $X_{ns}^{+}(p)$ for primes $p > 11$, we show that ${\rm Gal}(\mathbb{Q}(E[n])/\mathbb{Q})$ nilpotent implies that $n$ is a power of $2$ or $n \in \{ 3, 5, 6, 7, 15, 21 \}$.

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arXiv2024

Minimal Subgroups of ${\rm GL}_2(\mathbb{Z}_{S})$

Let $E$ be an elliptic curve over a number field $L$ and for a finite set $S$ of primes, let $\rho_{E,S} : {\rm Gal}(\overline{L}/L) \to {\rm GL}_{2}(\mathbb{Z}_{S})$ be the $S$-adic Galois representation. If $L \cap \mathbb{Q}(\zeta_{n}) = \mathbb{Q}$ for all positive integers $n$ whose prime factors are in $S$, then $\det \rho_{E,S} : {\rm Gal}(\overline{L}/L) \to \mathbb{Z}_{S}^{\times}$ is surjective. We say that a finite index subgroup $H \subseteq {\rm GL}_{2}(\mathbb{Z}_{S})$ is minimal if $\det : H \to \mathbb{Z}_{S}^{\times}$ is surjective, but $\det : K \to \mathbb{Z}_{S}^{\times}$ is not surjective for any proper closed subgroup $K$ of $H$. We show that there are no minimal subgroups of ${\rm GL}_{2}(\mathbb{Z}_{S})$ unless $S = \{ 2 \}$, while minimal subgroups of ${\rm GL}_{2}(\mathbb{Z}_{2})$ are plentiful. We give models for all the genus $0$ modular curves associated to minimal subgroups of ${\rm GL}_{2}(\mathbb{Z}_{2})$, and construct an infinite family of elliptic curves over imaginary quadratic fields with bad reduction only at $2$ and with minimal $2$-adic image.

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arXiv2023

Densities of integer sets represented by quadratic forms

Let $f(t_1,\ldots,t_n)$ be a nondegenerate integral quadratic form. We analyze the asymptotic behavior of the function $D_f(X)$, the number of integers of absolute value up to $X$ represented by $f$. When $f$ is isotropic or $n$ is at least $3$, we show that there is a $\delta(f) \in \mathbb{Q} \cap (0,1)$ such that $D_f(X) \sim \delta(f) X$ and call $\delta(f)$ the density of $f$. We consider the inverse problem of which densities arise. Our main technical tool is a Near Hasse Principle: a quadratic form may fail to represent infinitely many integers that it locally represents, but this set of exceptions has density $0$ within the set of locally represented integers.

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arXiv2022

Quaternary quadratic forms with prime discriminant

Let $Q$ be a positive-definite quaternary quadratic form with prime discriminant. We give an explicit lower bound on the number of representations of a positive integer $n$ by $Q$. This problem is connected with deriving an upper bound on the Petersson norm $\langle C, C \rangle$ of the cuspidal part of the theta series of $Q$. We derive an upper bound on $\langle C, C \rangle$ that depends on the smallest positive integer not represented by the dual form $Q^{*}$. In addition, we give a non-trivial upper bound on the sum of the integers $n$ excepted by $Q$.

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arXiv2022

Rational points on $x^{3} + x^{2} y^{2} + y^{3} = k$

We study the problem of determining, given an integer $k$, the rational solutions to $C_{k} : x^{3}z + x^{2} y^{2} + y^{3}z = kz^{4}$. For $k \ne 0$, the curve $C_{k}$ has genus $3$ and there are maps from $C_{k}$ to three elliptic curves $E_{1,k}$, $E_{2,k}$, $E_{3,k}$. We explicitly determine the rational points on $C_{k}$ under the assumption that one of these elliptic curves has rank zero. We discuss the challenges involved in extending our result to handle all $k \in \mathbb{Q}$.

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arXiv2021

$\ell$-adic images of Galois for elliptic curves over $\mathbb{Q}$

We discuss the $\ell$-adic case of Mazur's "Program B" over $\mathbb{Q}$, the problem of classifying the possible images of $\ell$-adic Galois representations attached to elliptic curves $E$ over $\mathbb{Q}$, equivalently, classifying the rational points on the corresponding modular curves. The primes $\ell=2$ and $\ell\ge 13$ are addressed by prior work, so we focus on the remaining primes $\ell = 3, 5, 7, 11$. For each of these $\ell$, we compute the directed graph of arithmetically maximal $\ell$-power level modular curves $X_H$, compute explicit equations for all but three of them, and classify the rational points on all of them except $X_{{\rm ns}}^{+}(N)$, for $N = 27, 25, 49, 121$, and two level $49$ curves of genus $9$ whose Jacobians have analytic rank $9$. Aside from the $\ell$-adic images that are known to arise for infinitely many $\bar{\mathbb{Q}}$-isomorphism classes of elliptic curves $E/\mathbb{Q}$, we find only 22 exceptional images that arise for any prime $\ell$ and any $E/\mathbb{Q}$ without complex multiplication; these exceptional images are realized by 20 non-CM rational $j$-invariants. We conjecture that this list of 22 exceptional images is complete and show that any counterexamples must arise from unexpected rational points on $X_{\rm ns}^+(\ell)$ with $\ell\ge 19$, or one of the six modular curves noted above. This yields a very efficient algorithm to compute the $\ell$-adic images of Galois for any elliptic curve over $\mathbb{Q}$. In an appendix with John Voight we generalize Ribet's observation that simple abelian varieties attached to newforms on $\Gamma_1(N)$ are of ${\rm GL}_2$-type; this extends Kolyvagin's theorem that analytic rank zero implies algebraic rank zero to isogeny factors of the Jacobian of $X_H$.

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arXiv2021

Integers that are sums of two rational sixth powers

We prove that $164634913$ is the smallest positive integer that is a sum of two rational sixth powers but not a sum of two integer sixth powers. If $C_{k}$ is the curve $x^{6} + y^{6} = k$, we use the existence of morphisms from $C_{k}$ to elliptic curves, together with the Mordell-Weil sieve, to rule out the existence of rational points on $C_{k}$ for various $k$.

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arXiv2020

Odd degree isolated points on $X_1(N)$ with rational $j$-invariant

Let $C$ be a curve defined over a number field $k$. We say a closed point $x\in C$ of degree $d$ is isolated if it does not belong to an infinite family of degree $d$ points parametrized by the projective line or a positive rank abelian subvariety of the curve's Jacobian. Building on work of Bourdon, Ejder, Liu, Odumodu, and Viray, we characterize elliptic curves with rational $j$-invariant which give rise to an isolated point of odd degree on $X_1(N)/\mathbb{Q}$ for some positive integer $N$.

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arXiv2019

The number of representations of $n$ as a growing number of squares

Let $r_{k}(n)$ denote the number of representations of the integer $n$ as a sum of $k$ squares. In this paper, we give an asymptotic for $r_{k}(n)$ when $n$ grows linearly with $k$. As a special case, we find that \[ r_{n}(n) \sim \frac{B \cdot A^{n}}{\sqrt{n}}, \] with $B \approx 0.2821$ and $A \approx 4.133$.

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arXiv2019

Uniform bounds on the image of the arboreal Galois representations attached to non-CM elliptic curves

Let $\ell$ be a prime number and let $F$ be a number field and $E/F$ a non-CM elliptic curve with a point $\alpha \in E(F)$ of infinite order. Attached to the pair $(E,\alpha)$ is the $\ell$-adic arboreal Galois representation $\omega_{E,\alpha,\ell^{\infty}} : {\rm Gal}(\overline{F}/F) \to \mathbb{Z}_{\ell}^{2} \rtimes {\rm GL}_{2}(\mathbb{Z}_{\ell})$ describing the action of ${\rm Gal}(\overline{F}/F)$ on points $\beta_{n}$ so that $\ell^{n} \beta_{n} = \alpha$. We give an explicit bound on the index of the image of $\omega_{E,\alpha,\ell^{\infty}}$ depending on how $\ell$-divisible the point $\alpha$ is, and the image of the ordinary $\ell$-adic Galois representation. The image of $\omega_{E,\alpha,\ell^{\infty}}$ is connected with the density of primes $\mathfrak{p}$ for which $\alpha \in E(\mathbb{F}_{\mathfrak{p}})$ has order coprime to $\ell$.

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arXiv2018

$p$-adic quotient sets II: quadratic forms

For $A \subseteq \{1,2,\ldots\}$, we consider $R(A) = \{a/a' : a,a' \in A\}$. If $A$ is the set of nonzero values assumed by a quadratic form, when is $R(A)$ dense in the $p$-adic numbers? We show that for a binary quadratic form $Q$, $R(A)$ is dense in $\mathbb{Q}_{p}$ if and only if the discriminant of $Q$ is a nonzero square in $\mathbb{Q}_{p}$, and for a quadratic form in at least three variables, $R(A)$ is always dense in $\mathbb{Q}_{p}$. This answers a question posed by several authors in 2017.

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arXiv2018

The density of odd order reductions for elliptic curves with a rational point of order 2

Suppose that $E/\mathbb{Q}$ is an elliptic curve with a rational point $T$ of order $2$ and $\alpha \in E(\mathbb{Q})$ is a point of infinite order. We consider the problem of determining the density of primes $p$ for which $\alpha \in E(\mathbb{F}_{p})$ has odd order. This density is determined by the image of the arboreal Galois representation $\tau_{E,2^{k}} : {\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to {\rm AGL}_{2}(\mathbb{Z}/2^{k}\mathbb{Z})$. Assuming that $\alpha$ is primitive (that is, neither $\alpha$ nor $\alpha + T$ is twice a point over $\mathbb{Q}$) and that the image of the ordinary mod $2^{k}$ Galois representation is as large as possible (subject to $E$ having a rational point of order $2$), we determine that there are $63$ possibilities for the image of $\tau_{E,2^{k}}$. As a consequence, the density of primes $p$ for which the order of $\alpha$ is odd is between $1/14$ and $89/168$.

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arXiv2018

Integers represented by positive-definite quadratic forms and Petersson inner products

Let $Q$ be a positive-definite quaternary quadratic form with integer coefficients. We study the problem of giving bounds on the largest positive integer $n$ that is locally represented by $Q$ but not represented. Assuming that $n$ is relatively prime to $D(Q)$, the determinant of the Gram matrix of $Q$, we show that $n$ is represented provided that \[ n \gg \max \{ N(Q)^{3/2 + \epsilon} D(Q)^{5/4 + \epsilon}, N(Q)^{2 + \epsilon} D(Q)^{1 + \epsilon} \}. \] Here $N(Q)$ is the level of $Q$. We give three other bounds that hold under successively weaker local conditions on $n$. These results are proven by bounding the Petersson norm of the cuspidal part of the theta series, which is accomplished using an explicit formula for the Weil representation due to Scheithauer.

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arXiv2017

An Elliptic Curve Analogue to the Fermat Numbers

The Fermat numbers have many notable properties, including order universality, coprimality, and definition by a recurrence relation. We use arbitrary elliptic curves and rational points of infinite order to generate sequences that are analogous to the Fermat numbers. We demonstrate that these sequences have many of the same properties as the Fermat numbers, and we discuss results about the prime factors of sequences generated by specific curves and points.

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arXiv2016

Quadratic forms representing all integers coprime to 3

Following Bhargava and Hanke's celebrated 290-theorem, we prove a universality theorem for all positive-definite integer-valued quadratic forms that represent all positive integers coprime to $3$. In particular, if a positive-definite quadratic form represents all positive integers coprime to $3$ and $\leq 290$, then it represents all positive integers coprime to $3$. We use similar methods to those used by Rouse to prove (assuming GRH) that a positive-definite quadratic form representing every odd integer between $1$ and $451$ represents all positive odd integers.

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